%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP130-4.003 : TPTP v9.0.0. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:07:11 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.38s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.11 % Problem : GRP130-4.003 : TPTP v9.0.0. Bugfixed v1.2.1.
% 0.07/0.11 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n009.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Thu Jun 19 15:43:48 EDT 2025
% 0.11/0.32 % CPUTime :
% 0.19/0.38 % SZS status Unsatisfiable
% 0.19/0.38 % SZS output start Proof
% 0.19/0.38 tff(product_type, type, (
% 0.19/0.38 product: ( $i * $i * $i ) > $o)).
% 0.19/0.38 tff(e_1_type, type, (
% 0.19/0.38 e_1: $i)).
% 0.19/0.38 tff(e_2_type, type, (
% 0.19/0.38 e_2: $i)).
% 0.19/0.38 tff(equalish_type, type, (
% 0.19/0.38 equalish: ( $i * $i ) > $o)).
% 0.19/0.38 tff(e_3_type, type, (
% 0.19/0.38 e_3: $i)).
% 0.19/0.38 tff(group_element_type, type, (
% 0.19/0.38 group_element: $i > $o)).
% 0.19/0.38 tff(1,assumption,(product(e_1, e_2, e_1)), introduced(assumption)).
% 0.19/0.38 tff(2,plain,
% 0.19/0.38 (^[W: $i, Z: $i, Y: $i, X: $i] : refl(((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y))) <=> ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(3,plain,
% 0.19/0.38 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[2])).
% 0.19/0.38 tff(4,plain,
% 0.19/0.38 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(5,plain,
% 0.19/0.38 (^[W: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product(X, W, Y)) | (~product(X, Z, Y))) <=> ((~product(X, W, Y)) | (~product(X, Z, Y)))), ((((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z)) <=> (((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z)))), rewrite((((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z)) <=> ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))), ((((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z)) <=> ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(6,plain,
% 0.19/0.38 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[5])).
% 0.19/0.38 tff(7,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, W, Y)) | (~product(X, Z, Y))) | equalish(W, Z))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','product_right_cancellation')).
% 0.19/0.38 tff(8,plain,
% 0.19/0.38 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[7, 6])).
% 0.19/0.38 tff(9,plain,
% 0.19/0.38 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[8, 4])).
% 0.19/0.38 tff(10,plain,(
% 0.19/0.38 ![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[9])).
% 0.19/0.38 tff(11,plain,
% 0.19/0.38 (![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[10, 3])).
% 0.19/0.38 tff(12,plain,
% 0.19/0.38 ((~equalish(e_2, e_1)) <=> (~equalish(e_2, e_1))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(13,axiom,(~equalish(e_2, e_1)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_2_is_not_e_1')).
% 0.19/0.38 tff(14,plain,
% 0.19/0.38 (~equalish(e_2, e_1)),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.19/0.38 tff(15,plain,
% 0.19/0.38 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(16,plain,
% 0.19/0.38 (((~product(e_1, e_2, e_1)) | equalish(e_2, e_1) | (~product(e_1, e_1, e_1))) <=> (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(17,plain,
% 0.19/0.38 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | ((~product(e_1, e_2, e_1)) | equalish(e_2, e_1) | (~product(e_1, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | (equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1))))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[16])).
% 0.19/0.38 tff(18,plain,
% 0.19/0.38 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | ((~product(e_1, e_2, e_1)) | equalish(e_2, e_1) | (~product(e_1, e_1, e_1)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1)))),
% 0.19/0.38 inference(transitivity,[status(thm)],[17, 15])).
% 0.19/0.38 tff(19,plain,
% 0.19/0.38 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | ((~product(e_1, e_2, e_1)) | equalish(e_2, e_1) | (~product(e_1, e_1, e_1)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(20,plain,
% 0.19/0.38 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | equalish(e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[19, 18])).
% 0.19/0.38 tff(21,plain,
% 0.19/0.38 ((~product(e_1, e_1, e_1)) | (~product(e_1, e_2, e_1))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[20, 14, 11])).
% 0.19/0.38 tff(22,plain,
% 0.19/0.38 (~product(e_1, e_1, e_1)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[21, 1])).
% 0.19/0.38 tff(23,plain,
% 0.19/0.38 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : refl((product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1))) <=> (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(24,plain,
% 0.19/0.38 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[23])).
% 0.19/0.38 tff(25,plain,
% 0.19/0.38 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(26,plain,
% 0.19/0.38 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : trans(monotonicity(rewrite((product(X, Z1, Z2) | (~product(Z2, Y, X))) <=> (product(X, Z1, Z2) | (~product(Z2, Y, X)))), (((product(X, Z1, Z2) | (~product(Z2, Y, X))) | (~product(X, Y, Z1))) <=> ((product(X, Z1, Z2) | (~product(Z2, Y, X))) | (~product(X, Y, Z1))))), rewrite(((product(X, Z1, Z2) | (~product(Z2, Y, X))) | (~product(X, Y, Z1))) <=> (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))), (((product(X, Z1, Z2) | (~product(Z2, Y, X))) | (~product(X, Y, Z1))) <=> (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(27,plain,
% 0.19/0.38 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((product(X, Z1, Z2) | (~product(Z2, Y, X))) | (~product(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[26])).
% 0.19/0.38 tff(28,axiom,(![Y: $i, Z1: $i, X: $i, Z2: $i] : ((product(X, Z1, Z2) | (~product(Z2, Y, X))) | (~product(X, Y, Z1)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','qg3_2')).
% 0.19/0.38 tff(29,plain,
% 0.19/0.38 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.19/0.38 tff(30,plain,
% 0.19/0.38 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[29, 25])).
% 0.19/0.38 tff(31,plain,(
% 0.19/0.38 ![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[30])).
% 0.19/0.38 tff(32,plain,
% 0.19/0.38 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[31, 24])).
% 0.19/0.38 tff(33,plain,
% 0.19/0.38 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_1) | (~product(e_1, e_2, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | product(e_1, e_1, e_1) | (~product(e_1, e_2, e_1)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(34,plain,
% 0.19/0.39 ((product(e_1, e_1, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_2, e_1))) <=> (product(e_1, e_1, e_1) | (~product(e_1, e_2, e_1)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(35,plain,
% 0.19/0.39 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_2, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_1) | (~product(e_1, e_2, e_1))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[34])).
% 0.19/0.39 tff(36,plain,
% 0.19/0.39 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_2, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | product(e_1, e_1, e_1) | (~product(e_1, e_2, e_1)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[35, 33])).
% 0.19/0.39 tff(37,plain,
% 0.19/0.39 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_1) | (~product(e_1, e_2, e_1)) | (~product(e_1, e_2, e_1)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(38,plain,
% 0.19/0.39 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | product(e_1, e_1, e_1) | (~product(e_1, e_2, e_1))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.19/0.39 tff(39,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[38, 32, 22, 1])).
% 0.19/0.39 tff(40,plain,(~product(e_1, e_2, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(41,assumption,(product(e_1, e_3, e_1)), introduced(assumption)).
% 0.19/0.39 tff(42,plain,
% 0.19/0.39 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : refl(((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1)) <=> ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(43,plain,
% 0.19/0.39 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1)) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[42])).
% 0.19/0.39 tff(44,plain,
% 0.19/0.39 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1)) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(45,plain,
% 0.19/0.39 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : trans(monotonicity(rewrite((product(X, Y, Z1) | (~product(Z2, Y, X))) <=> ((~product(Z2, Y, X)) | product(X, Y, Z1))), (((product(X, Y, Z1) | (~product(Z2, Y, X))) | (~product(X, Z1, Z2))) <=> (((~product(Z2, Y, X)) | product(X, Y, Z1)) | (~product(X, Z1, Z2))))), rewrite((((~product(Z2, Y, X)) | product(X, Y, Z1)) | (~product(X, Z1, Z2))) <=> ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))), (((product(X, Y, Z1) | (~product(Z2, Y, X))) | (~product(X, Z1, Z2))) <=> ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(46,plain,
% 0.19/0.39 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((product(X, Y, Z1) | (~product(Z2, Y, X))) | (~product(X, Z1, Z2))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[45])).
% 0.19/0.39 tff(47,axiom,(![Y: $i, Z1: $i, X: $i, Z2: $i] : ((product(X, Y, Z1) | (~product(Z2, Y, X))) | (~product(X, Z1, Z2)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','qg3_1')).
% 0.19/0.39 tff(48,plain,
% 0.19/0.39 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.19/0.39 tff(49,plain,
% 0.19/0.39 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[48, 44])).
% 0.19/0.39 tff(50,plain,(
% 0.19/0.39 ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))),
% 0.19/0.39 inference(skolemize,[status(sab)],[49])).
% 0.19/0.39 tff(51,plain,
% 0.19/0.39 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[50, 43])).
% 0.19/0.39 tff(52,plain,
% 0.19/0.39 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | (product(e_1, e_3, e_3) | (~product(e_1, e_3, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | product(e_1, e_3, e_3) | (~product(e_1, e_3, e_1)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(53,plain,
% 0.19/0.39 (((~product(e_1, e_3, e_1)) | (~product(e_1, e_3, e_1)) | product(e_1, e_3, e_3)) <=> (product(e_1, e_3, e_3) | (~product(e_1, e_3, e_1)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(54,plain,
% 0.19/0.39 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | ((~product(e_1, e_3, e_1)) | (~product(e_1, e_3, e_1)) | product(e_1, e_3, e_3))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | (product(e_1, e_3, e_3) | (~product(e_1, e_3, e_1))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[53])).
% 0.19/0.39 tff(55,plain,
% 0.19/0.39 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | ((~product(e_1, e_3, e_1)) | (~product(e_1, e_3, e_1)) | product(e_1, e_3, e_3))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | product(e_1, e_3, e_3) | (~product(e_1, e_3, e_1)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[54, 52])).
% 0.19/0.39 tff(56,plain,
% 0.19/0.39 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | ((~product(e_1, e_3, e_1)) | (~product(e_1, e_3, e_1)) | product(e_1, e_3, e_3))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(57,plain,
% 0.19/0.39 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | product(e_1, e_3, e_3) | (~product(e_1, e_3, e_1))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.19/0.39 tff(58,plain,
% 0.19/0.39 (product(e_1, e_3, e_3)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[57, 51, 41])).
% 0.19/0.39 tff(59,plain,
% 0.19/0.39 (^[W: $i, Z: $i, Y: $i, X: $i] : refl((equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W))) <=> (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(60,plain,
% 0.19/0.39 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[59])).
% 0.19/0.39 tff(61,plain,
% 0.19/0.39 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W))) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(62,plain,
% 0.19/0.39 (^[W: $i, Z: $i, Y: $i, X: $i] : trans(monotonicity(rewrite(((~product(X, Y, W)) | (~product(X, Y, Z))) <=> ((~product(X, Y, Z)) | (~product(X, Y, W)))), ((((~product(X, Y, W)) | (~product(X, Y, Z))) | equalish(W, Z)) <=> (((~product(X, Y, Z)) | (~product(X, Y, W))) | equalish(W, Z)))), rewrite((((~product(X, Y, Z)) | (~product(X, Y, W))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))), ((((~product(X, Y, W)) | (~product(X, Y, Z))) | equalish(W, Z)) <=> (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(63,plain,
% 0.19/0.39 (![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, W)) | (~product(X, Y, Z))) | equalish(W, Z)) <=> ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[62])).
% 0.19/0.39 tff(64,axiom,(![W: $i, Z: $i, Y: $i, X: $i] : (((~product(X, Y, W)) | (~product(X, Y, Z))) | equalish(W, Z))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','product_total_function2')).
% 0.19/0.39 tff(65,plain,
% 0.19/0.39 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.19/0.39 tff(66,plain,
% 0.19/0.39 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[65, 61])).
% 0.19/0.39 tff(67,plain,(
% 0.19/0.39 ![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[66])).
% 0.19/0.39 tff(68,plain,
% 0.19/0.39 (![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[67, 60])).
% 0.19/0.39 tff(69,plain,
% 0.19/0.39 ((~equalish(e_3, e_1)) <=> (~equalish(e_3, e_1))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(70,axiom,(~equalish(e_3, e_1)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_3_is_not_e_1')).
% 0.19/0.39 tff(71,plain,
% 0.19/0.39 (~equalish(e_3, e_1)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.19/0.39 tff(72,plain,
% 0.19/0.39 (((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_3, e_1) | (~product(e_1, e_3, e_1)) | (~product(e_1, e_3, e_3)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_3, e_1) | (~product(e_1, e_3, e_1)) | (~product(e_1, e_3, e_3)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(73,plain,
% 0.19/0.39 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | (equalish(e_3, e_1) | (~product(e_1, e_3, e_1)) | (~product(e_1, e_3, e_3)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(74,plain,
% 0.19/0.39 ((~![W: $i, Z: $i, Y: $i, X: $i] : (equalish(W, Z) | (~product(X, Y, Z)) | (~product(X, Y, W)))) | equalish(e_3, e_1) | (~product(e_1, e_3, e_1)) | (~product(e_1, e_3, e_3))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.19/0.39 tff(75,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[74, 71, 68, 41, 58])).
% 0.19/0.39 tff(76,plain,(~product(e_1, e_3, e_1)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(77,plain,
% 0.19/0.39 (group_element(e_1) <=> group_element(e_1)),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(78,axiom,(group_element(e_1)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','element_1')).
% 0.19/0.39 tff(79,plain,
% 0.19/0.39 (group_element(e_1)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.19/0.39 tff(80,plain,
% 0.19/0.39 (^[Y: $i, X: $i] : refl(((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(81,plain,
% 0.19/0.39 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[80])).
% 0.19/0.39 tff(82,plain,
% 0.19/0.39 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(83,plain,
% 0.19/0.39 (^[Y: $i, X: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_1, Y))), (((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, e_1, Y)) | product(X, e_2, Y)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, e_1, Y)) | product(X, e_2, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_2, Y) | product(X, e_1, Y))), (((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_2, Y) | product(X, e_1, Y)))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, e_2, Y) | product(X, e_1, Y)) | product(X, e_3, Y)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, e_2, Y) | product(X, e_1, Y)) | product(X, e_3, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(84,plain,
% 0.19/0.40 (![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[83])).
% 0.19/0.40 tff(85,axiom,(![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(X, e_1, Y)) | product(X, e_2, Y)) | product(X, e_3, Y))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','column_surjectivity')).
% 0.19/0.40 tff(86,plain,
% 0.19/0.40 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.19/0.40 tff(87,plain,
% 0.19/0.40 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[86, 82])).
% 0.19/0.40 tff(88,plain,(
% 0.19/0.40 ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.40 inference(skolemize,[status(sab)],[87])).
% 0.19/0.40 tff(89,plain,
% 0.19/0.40 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[88, 81])).
% 0.19/0.40 tff(90,plain,
% 0.19/0.40 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | product(e_1, e_2, e_1) | product(e_1, e_3, e_1) | product(e_1, e_1, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (~group_element(e_1)) | product(e_1, e_2, e_1) | product(e_1, e_3, e_1) | product(e_1, e_1, e_1))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(91,plain,
% 0.19/0.40 (((~group_element(e_1)) | (~group_element(e_1)) | product(e_1, e_3, e_1) | product(e_1, e_2, e_1) | product(e_1, e_1, e_1)) <=> ((~group_element(e_1)) | product(e_1, e_2, e_1) | product(e_1, e_3, e_1) | product(e_1, e_1, e_1))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(92,plain,
% 0.19/0.40 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | (~group_element(e_1)) | product(e_1, e_3, e_1) | product(e_1, e_2, e_1) | product(e_1, e_1, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | product(e_1, e_2, e_1) | product(e_1, e_3, e_1) | product(e_1, e_1, e_1)))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[91])).
% 0.19/0.40 tff(93,plain,
% 0.19/0.40 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | (~group_element(e_1)) | product(e_1, e_3, e_1) | product(e_1, e_2, e_1) | product(e_1, e_1, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (~group_element(e_1)) | product(e_1, e_2, e_1) | product(e_1, e_3, e_1) | product(e_1, e_1, e_1))),
% 0.19/0.40 inference(transitivity,[status(thm)],[92, 90])).
% 0.19/0.40 tff(94,plain,
% 0.19/0.40 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | ((~group_element(e_1)) | (~group_element(e_1)) | product(e_1, e_3, e_1) | product(e_1, e_2, e_1) | product(e_1, e_1, e_1))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(95,plain,
% 0.19/0.40 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, e_3, Y) | product(X, e_2, Y) | product(X, e_1, Y))) | (~group_element(e_1)) | product(e_1, e_2, e_1) | product(e_1, e_3, e_1) | product(e_1, e_1, e_1)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[94, 93])).
% 0.19/0.40 tff(96,plain,
% 0.19/0.40 (product(e_1, e_3, e_1) | product(e_1, e_1, e_1)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[95, 89, 79, 40])).
% 0.19/0.40 tff(97,plain,
% 0.19/0.40 (product(e_1, e_1, e_1)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[96, 76])).
% 0.19/0.40 tff(98,plain,
% 0.19/0.40 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | ((~product(e_1, e_1, e_2)) | (~product(e_1, e_1, e_1)) | product(e_1, e_2, e_1))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_1, e_1)) | product(e_1, e_2, e_1))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(99,plain,
% 0.19/0.40 ((product(e_1, e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_1, e_2))) <=> ((~product(e_1, e_1, e_2)) | (~product(e_1, e_1, e_1)) | product(e_1, e_2, e_1))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(100,plain,
% 0.19/0.40 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (product(e_1, e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_1, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | ((~product(e_1, e_1, e_2)) | (~product(e_1, e_1, e_1)) | product(e_1, e_2, e_1)))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[99])).
% 0.19/0.40 tff(101,plain,
% 0.19/0.40 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (product(e_1, e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_1, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_1, e_1)) | product(e_1, e_2, e_1))),
% 0.19/0.40 inference(transitivity,[status(thm)],[100, 98])).
% 0.19/0.40 tff(102,plain,
% 0.19/0.40 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (product(e_1, e_2, e_1) | (~product(e_1, e_1, e_1)) | (~product(e_1, e_1, e_2)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(103,plain,
% 0.19/0.40 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (~product(e_1, e_1, e_2)) | (~product(e_1, e_1, e_1)) | product(e_1, e_2, e_1)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.19/0.40 tff(104,plain,
% 0.19/0.40 (~product(e_1, e_1, e_2)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[103, 32, 97, 40])).
% 0.19/0.40 tff(105,assumption,(~product(e_3, e_2, e_2)), introduced(assumption)).
% 0.19/0.40 tff(106,plain,
% 0.19/0.40 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | (product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(107,plain,
% 0.19/0.40 (((~product(e_3, e_2, e_3)) | (~product(e_3, e_2, e_3)) | product(e_3, e_2, e_2)) <=> (product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(108,plain,
% 0.19/0.40 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | ((~product(e_3, e_2, e_3)) | (~product(e_3, e_2, e_3)) | product(e_3, e_2, e_2))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | (product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[107])).
% 0.19/0.40 tff(109,plain,
% 0.19/0.40 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | ((~product(e_3, e_2, e_3)) | (~product(e_3, e_2, e_3)) | product(e_3, e_2, e_2))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3)))),
% 0.19/0.40 inference(transitivity,[status(thm)],[108, 106])).
% 0.19/0.40 tff(110,plain,
% 0.19/0.40 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | ((~product(e_3, e_2, e_3)) | (~product(e_3, e_2, e_3)) | product(e_3, e_2, e_2))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(111,plain,
% 0.19/0.40 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[110, 109])).
% 0.19/0.40 tff(112,plain,
% 0.19/0.40 (product(e_3, e_2, e_2) | (~product(e_3, e_2, e_3))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[111, 51])).
% 0.19/0.40 tff(113,plain,
% 0.19/0.40 (~product(e_3, e_2, e_3)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[112, 105])).
% 0.19/0.40 tff(114,plain,
% 0.19/0.40 (^[Y: $i, X: $i] : refl(((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(115,plain,
% 0.19/0.40 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[114])).
% 0.19/0.40 tff(116,plain,
% 0.19/0.40 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(117,plain,
% 0.19/0.40 (^[Y: $i, X: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_1))), (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_1)) | product(X, Y, e_2)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_2) | product(X, Y, e_1))), (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_2) | product(X, Y, e_1)))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> (((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_2) | product(X, Y, e_1)) | product(X, Y, e_3)))), rewrite((((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_2) | product(X, Y, e_1)) | product(X, Y, e_3)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))), ((((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(118,plain,
% 0.19/0.40 (![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3)) <=> ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[117])).
% 0.19/0.40 tff(119,axiom,(![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(X, Y, e_1)) | product(X, Y, e_2)) | product(X, Y, e_3))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','product_total_function1')).
% 0.19/0.41 tff(120,plain,
% 0.19/0.41 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[119, 118])).
% 0.19/0.41 tff(121,plain,
% 0.19/0.41 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[120, 116])).
% 0.19/0.41 tff(122,plain,(
% 0.19/0.41 ![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.41 inference(skolemize,[status(sab)],[121])).
% 0.19/0.41 tff(123,plain,
% 0.19/0.41 (![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[122, 115])).
% 0.19/0.41 tff(124,plain,
% 0.19/0.41 (group_element(e_3) <=> group_element(e_3)),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(125,axiom,(group_element(e_3)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','element_3')).
% 0.19/0.41 tff(126,plain,
% 0.19/0.41 (group_element(e_3)),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.19/0.41 tff(127,plain,
% 0.19/0.41 (group_element(e_2) <=> group_element(e_2)),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(128,axiom,(group_element(e_2)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','element_2')).
% 0.19/0.41 tff(129,plain,
% 0.19/0.41 (group_element(e_2)),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[128, 127])).
% 0.19/0.41 tff(130,plain,
% 0.19/0.41 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_1))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(131,plain,
% 0.19/0.41 (((~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | product(e_3, e_2, e_1)) <=> (product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_1))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(132,plain,
% 0.19/0.41 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | product(e_3, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | (product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_1)))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[131])).
% 0.19/0.41 tff(133,plain,
% 0.19/0.41 (((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | product(e_3, e_2, e_1))) <=> ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_1))),
% 0.19/0.41 inference(transitivity,[status(thm)],[132, 130])).
% 0.19/0.41 tff(134,plain,
% 0.19/0.41 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | ((~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | product(e_3, e_2, e_1))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(135,plain,
% 0.19/0.41 ((~![Y: $i, X: $i] : ((~group_element(Y)) | (~group_element(X)) | product(X, Y, e_3) | product(X, Y, e_2) | product(X, Y, e_1))) | product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_3, e_2, e_1)),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[134, 133])).
% 0.19/0.41 tff(136,plain,
% 0.19/0.41 (product(e_3, e_2, e_3) | product(e_3, e_2, e_2) | product(e_3, e_2, e_1)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[135, 129, 126, 123])).
% 0.19/0.41 tff(137,plain,
% 0.19/0.41 (product(e_3, e_2, e_1)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[136, 113, 105])).
% 0.19/0.41 tff(138,plain,
% 0.19/0.41 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | (product(e_3, e_2, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | product(e_3, e_2, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(139,plain,
% 0.19/0.41 (((~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)) | product(e_3, e_2, e_2)) <=> (product(e_3, e_2, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(140,plain,
% 0.19/0.41 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | ((~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)) | product(e_3, e_2, e_2))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | (product(e_3, e_2, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[139])).
% 0.19/0.41 tff(141,plain,
% 0.19/0.41 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | ((~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)) | product(e_3, e_2, e_2))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | product(e_3, e_2, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)))),
% 0.19/0.41 inference(transitivity,[status(thm)],[140, 138])).
% 0.19/0.41 tff(142,plain,
% 0.19/0.41 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | ((~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3)) | product(e_3, e_2, e_2))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(143,plain,
% 0.19/0.41 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | product(e_3, e_2, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[142, 141])).
% 0.19/0.41 tff(144,plain,
% 0.19/0.41 (product(e_3, e_2, e_2) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_3))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[143, 51])).
% 0.19/0.41 tff(145,plain,
% 0.19/0.41 (~product(e_1, e_2, e_3)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[144, 137, 105])).
% 0.19/0.41 tff(146,plain,
% 0.19/0.41 (^[Y: $i, X: $i] : refl((product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) <=> (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(147,plain,
% 0.19/0.41 (![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) <=> ![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[146])).
% 0.19/0.41 tff(148,plain,
% 0.19/0.41 (![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) <=> ![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(149,plain,
% 0.19/0.41 (^[Y: $i, X: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) <=> (product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))), (((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) <=> ((product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_2, X, Y)))), rewrite(((product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_2, X, Y)) <=> (product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))), (((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) <=> (product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))))), ((((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y)) <=> ((product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_3, X, Y)))), rewrite(((product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X))) | product(e_3, X, Y)) <=> (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))), ((((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y)) <=> (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(150,plain,
% 0.19/0.41 (![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y)) <=> ![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[149])).
% 0.19/0.41 tff(151,axiom,(![Y: $i, X: $i] : (((((~group_element(X)) | (~group_element(Y))) | product(e_1, X, Y)) | product(e_2, X, Y)) | product(e_3, X, Y))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','row_surjectivity')).
% 0.19/0.41 tff(152,plain,
% 0.19/0.41 (![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[151, 150])).
% 0.19/0.41 tff(153,plain,
% 0.19/0.41 (![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[152, 148])).
% 0.19/0.41 tff(154,plain,(
% 0.19/0.41 ![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.41 inference(skolemize,[status(sab)],[153])).
% 0.19/0.41 tff(155,plain,
% 0.19/0.41 (![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[154, 147])).
% 0.19/0.41 tff(156,plain,
% 0.19/0.41 (((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_3) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_1, e_2, e_3) | product(e_2, e_2, e_3))) <=> ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_2, e_3) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_1, e_2, e_3) | product(e_2, e_2, e_3))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(157,plain,
% 0.19/0.41 ((product(e_3, e_2, e_3) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3) | (~group_element(e_3)) | (~group_element(e_2))) <=> (product(e_3, e_2, e_3) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_1, e_2, e_3) | product(e_2, e_2, e_3))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(158,plain,
% 0.19/0.41 (((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_3) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3) | (~group_element(e_3)) | (~group_element(e_2)))) <=> ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_3) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_1, e_2, e_3) | product(e_2, e_2, e_3)))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[157])).
% 0.19/0.41 tff(159,plain,
% 0.19/0.42 (((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_3) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3) | (~group_element(e_3)) | (~group_element(e_2)))) <=> ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_2, e_3) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_1, e_2, e_3) | product(e_2, e_2, e_3))),
% 0.19/0.42 inference(transitivity,[status(thm)],[158, 156])).
% 0.19/0.42 tff(160,plain,
% 0.19/0.42 ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_3) | product(e_2, e_2, e_3) | product(e_1, e_2, e_3) | (~group_element(e_3)) | (~group_element(e_2)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(161,plain,
% 0.19/0.42 ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_2, e_3) | (~group_element(e_2)) | (~group_element(e_3)) | product(e_1, e_2, e_3) | product(e_2, e_2, e_3)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[160, 159])).
% 0.19/0.42 tff(162,plain,
% 0.19/0.42 (product(e_2, e_2, e_3)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[161, 155, 129, 126, 113, 145])).
% 0.19/0.42 tff(163,plain,
% 0.19/0.42 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : refl(((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1))) <=> ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(164,plain,
% 0.19/0.42 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[163])).
% 0.19/0.42 tff(165,plain,
% 0.19/0.42 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1))) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(166,plain,
% 0.19/0.42 (^[Y: $i, Z1: $i, X: $i, Z2: $i] : trans(monotonicity(rewrite(((~product(X, Y, Z1)) | (~product(X, Z1, Z2))) <=> ((~product(X, Z1, Z2)) | (~product(X, Y, Z1)))), ((((~product(X, Y, Z1)) | (~product(X, Z1, Z2))) | product(Z2, Y, X)) <=> (((~product(X, Z1, Z2)) | (~product(X, Y, Z1))) | product(Z2, Y, X)))), rewrite((((~product(X, Z1, Z2)) | (~product(X, Y, Z1))) | product(Z2, Y, X)) <=> ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))), ((((~product(X, Y, Z1)) | (~product(X, Z1, Z2))) | product(Z2, Y, X)) <=> ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(167,plain,
% 0.19/0.42 (![Y: $i, Z1: $i, X: $i, Z2: $i] : (((~product(X, Y, Z1)) | (~product(X, Z1, Z2))) | product(Z2, Y, X)) <=> ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[166])).
% 0.19/0.42 tff(168,axiom,(![Y: $i, Z1: $i, X: $i, Z2: $i] : (((~product(X, Y, Z1)) | (~product(X, Z1, Z2))) | product(Z2, Y, X))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','qg3')).
% 0.19/0.42 tff(169,plain,
% 0.19/0.42 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[168, 167])).
% 0.19/0.42 tff(170,plain,
% 0.19/0.42 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[169, 165])).
% 0.19/0.42 tff(171,plain,(
% 0.19/0.42 ![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[170])).
% 0.19/0.42 tff(172,plain,
% 0.19/0.42 (![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[171, 164])).
% 0.19/0.42 tff(173,plain,
% 0.19/0.42 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | (product(e_3, e_2, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | product(e_3, e_2, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(174,plain,
% 0.19/0.42 (((~product(e_2, e_2, e_3)) | product(e_3, e_2, e_2) | (~product(e_2, e_2, e_2))) <=> (product(e_3, e_2, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(175,plain,
% 0.19/0.42 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | ((~product(e_2, e_2, e_3)) | product(e_3, e_2, e_2) | (~product(e_2, e_2, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | (product(e_3, e_2, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_2))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[174])).
% 0.19/0.42 tff(176,plain,
% 0.19/0.42 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | ((~product(e_2, e_2, e_3)) | product(e_3, e_2, e_2) | (~product(e_2, e_2, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | product(e_3, e_2, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_2)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[175, 173])).
% 0.19/0.42 tff(177,plain,
% 0.19/0.42 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | ((~product(e_2, e_2, e_3)) | product(e_3, e_2, e_2) | (~product(e_2, e_2, e_2)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(178,plain,
% 0.19/0.42 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | product(e_3, e_2, e_2) | (~product(e_2, e_2, e_3)) | (~product(e_2, e_2, e_2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[177, 176])).
% 0.19/0.42 tff(179,plain,
% 0.19/0.42 (~product(e_2, e_2, e_2)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[178, 172, 105, 162])).
% 0.19/0.42 tff(180,plain,
% 0.19/0.42 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | ((~product(e_3, e_2, e_1)) | product(e_1, e_2, e_3) | (~product(e_1, e_2, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (~product(e_3, e_2, e_1)) | product(e_1, e_2, e_3) | (~product(e_1, e_2, e_2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(181,plain,
% 0.19/0.42 ((product(e_1, e_2, e_3) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_2))) <=> ((~product(e_3, e_2, e_1)) | product(e_1, e_2, e_3) | (~product(e_1, e_2, e_2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(182,plain,
% 0.19/0.42 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (product(e_1, e_2, e_3) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | ((~product(e_3, e_2, e_1)) | product(e_1, e_2, e_3) | (~product(e_1, e_2, e_2))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[181])).
% 0.19/0.42 tff(183,plain,
% 0.19/0.42 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (product(e_1, e_2, e_3) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (~product(e_3, e_2, e_1)) | product(e_1, e_2, e_3) | (~product(e_1, e_2, e_2)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[182, 180])).
% 0.19/0.42 tff(184,plain,
% 0.19/0.42 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (product(e_1, e_2, e_3) | (~product(e_3, e_2, e_1)) | (~product(e_1, e_2, e_2)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(185,plain,
% 0.19/0.42 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : (product(X, Z1, Z2) | (~product(Z2, Y, X)) | (~product(X, Y, Z1)))) | (~product(e_3, e_2, e_1)) | product(e_1, e_2, e_3) | (~product(e_1, e_2, e_2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[184, 183])).
% 0.19/0.42 tff(186,plain,
% 0.19/0.42 (~product(e_1, e_2, e_2)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[185, 32, 137, 145])).
% 0.19/0.42 tff(187,plain,
% 0.19/0.42 (((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_2) | product(e_1, e_2, e_2))) <=> ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_2) | product(e_1, e_2, e_2))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(188,plain,
% 0.19/0.42 ((product(e_3, e_2, e_2) | product(e_2, e_2, e_2) | product(e_1, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_2))) <=> (product(e_3, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_2) | product(e_1, e_2, e_2))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(189,plain,
% 0.19/0.42 (((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_2) | product(e_2, e_2, e_2) | product(e_1, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_2)))) <=> ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_2) | product(e_1, e_2, e_2)))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[188])).
% 0.19/0.42 tff(190,plain,
% 0.19/0.42 (((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_2) | product(e_2, e_2, e_2) | product(e_1, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_2)))) <=> ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_2) | product(e_1, e_2, e_2))),
% 0.19/0.42 inference(transitivity,[status(thm)],[189, 187])).
% 0.19/0.42 tff(191,plain,
% 0.19/0.42 ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_2, e_2) | product(e_2, e_2, e_2) | product(e_1, e_2, e_2) | (~group_element(e_2)) | (~group_element(e_2)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(192,plain,
% 0.19/0.42 ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_3, e_2, e_2) | (~group_element(e_2)) | product(e_2, e_2, e_2) | product(e_1, e_2, e_2)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[191, 190])).
% 0.19/0.42 tff(193,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[192, 155, 129, 105, 186, 179])).
% 0.19/0.42 tff(194,plain,(product(e_3, e_2, e_2)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(195,assumption,(product(e_3, e_2, e_2)), introduced(assumption)).
% 0.19/0.42 tff(196,assumption,(product(e_3, e_1, e_2)), introduced(assumption)).
% 0.19/0.42 tff(197,plain,
% 0.19/0.42 ((~equalish(e_1, e_2)) <=> (~equalish(e_1, e_2))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(198,axiom,(~equalish(e_1, e_2)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','e_1_is_not_e_2')).
% 0.19/0.42 tff(199,plain,
% 0.19/0.42 (~equalish(e_1, e_2)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[198, 197])).
% 0.19/0.42 tff(200,plain,
% 0.19/0.42 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | (equalish(e_1, e_2) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | equalish(e_1, e_2) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_2, e_2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(201,plain,
% 0.19/0.42 (((~product(e_3, e_1, e_2)) | equalish(e_1, e_2) | (~product(e_3, e_2, e_2))) <=> (equalish(e_1, e_2) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_2, e_2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(202,plain,
% 0.19/0.42 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | ((~product(e_3, e_1, e_2)) | equalish(e_1, e_2) | (~product(e_3, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | (equalish(e_1, e_2) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_2, e_2))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[201])).
% 0.19/0.43 tff(203,plain,
% 0.19/0.43 (((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | ((~product(e_3, e_1, e_2)) | equalish(e_1, e_2) | (~product(e_3, e_2, e_2)))) <=> ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | equalish(e_1, e_2) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_2, e_2)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[202, 200])).
% 0.19/0.43 tff(204,plain,
% 0.19/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | ((~product(e_3, e_1, e_2)) | equalish(e_1, e_2) | (~product(e_3, e_2, e_2)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(205,plain,
% 0.19/0.43 ((~![W: $i, Z: $i, Y: $i, X: $i] : ((~product(X, W, Y)) | equalish(W, Z) | (~product(X, Z, Y)))) | equalish(e_1, e_2) | (~product(e_3, e_1, e_2)) | (~product(e_3, e_2, e_2))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[204, 203])).
% 0.19/0.43 tff(206,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[205, 199, 11, 196, 195])).
% 0.19/0.43 tff(207,plain,((~product(e_3, e_2, e_2)) | (~product(e_3, e_1, e_2))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.43 tff(208,plain,
% 0.19/0.43 (~product(e_3, e_1, e_2)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[207, 194])).
% 0.19/0.43 tff(209,plain,
% 0.19/0.43 (((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_1, e_2) | product(e_1, e_1, e_2) | (~group_element(e_1)) | (~group_element(e_2)) | product(e_3, e_1, e_2))) <=> ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_2, e_1, e_2) | product(e_1, e_1, e_2) | (~group_element(e_1)) | (~group_element(e_2)) | product(e_3, e_1, e_2))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(210,plain,
% 0.19/0.43 ((product(e_3, e_1, e_2) | product(e_2, e_1, e_2) | product(e_1, e_1, e_2) | (~group_element(e_2)) | (~group_element(e_1))) <=> (product(e_2, e_1, e_2) | product(e_1, e_1, e_2) | (~group_element(e_1)) | (~group_element(e_2)) | product(e_3, e_1, e_2))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(211,plain,
% 0.19/0.43 (((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_1, e_2) | product(e_2, e_1, e_2) | product(e_1, e_1, e_2) | (~group_element(e_2)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_2, e_1, e_2) | product(e_1, e_1, e_2) | (~group_element(e_1)) | (~group_element(e_2)) | product(e_3, e_1, e_2)))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[210])).
% 0.19/0.43 tff(212,plain,
% 0.19/0.43 (((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_1, e_2) | product(e_2, e_1, e_2) | product(e_1, e_1, e_2) | (~group_element(e_2)) | (~group_element(e_1)))) <=> ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_2, e_1, e_2) | product(e_1, e_1, e_2) | (~group_element(e_1)) | (~group_element(e_2)) | product(e_3, e_1, e_2))),
% 0.19/0.43 inference(transitivity,[status(thm)],[211, 209])).
% 0.19/0.43 tff(213,plain,
% 0.19/0.43 ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | (product(e_3, e_1, e_2) | product(e_2, e_1, e_2) | product(e_1, e_1, e_2) | (~group_element(e_2)) | (~group_element(e_1)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(214,plain,
% 0.19/0.43 ((~![Y: $i, X: $i] : (product(e_3, X, Y) | product(e_2, X, Y) | product(e_1, X, Y) | (~group_element(Y)) | (~group_element(X)))) | product(e_2, e_1, e_2) | product(e_1, e_1, e_2) | (~group_element(e_1)) | (~group_element(e_2)) | product(e_3, e_1, e_2)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[213, 212])).
% 0.19/0.43 tff(215,plain,
% 0.19/0.43 (product(e_2, e_1, e_2) | product(e_1, e_1, e_2) | product(e_3, e_1, e_2)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[214, 155, 79, 129])).
% 0.19/0.43 tff(216,plain,
% 0.19/0.43 (product(e_2, e_1, e_2) | product(e_1, e_1, e_2)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[215, 208])).
% 0.19/0.43 tff(217,plain,
% 0.19/0.43 (product(e_2, e_1, e_2)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[216, 104])).
% 0.19/0.43 tff(218,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | (product(e_2, e_1, e_1) | (~product(e_2, e_1, e_2)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | product(e_2, e_1, e_1) | (~product(e_2, e_1, e_2)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(219,plain,
% 0.19/0.43 (((~product(e_2, e_1, e_2)) | (~product(e_2, e_1, e_2)) | product(e_2, e_1, e_1)) <=> (product(e_2, e_1, e_1) | (~product(e_2, e_1, e_2)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(220,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | ((~product(e_2, e_1, e_2)) | (~product(e_2, e_1, e_2)) | product(e_2, e_1, e_1))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | (product(e_2, e_1, e_1) | (~product(e_2, e_1, e_2))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[219])).
% 0.19/0.43 tff(221,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | ((~product(e_2, e_1, e_2)) | (~product(e_2, e_1, e_2)) | product(e_2, e_1, e_1))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | product(e_2, e_1, e_1) | (~product(e_2, e_1, e_2)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[220, 218])).
% 0.19/0.43 tff(222,plain,
% 0.19/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | ((~product(e_2, e_1, e_2)) | (~product(e_2, e_1, e_2)) | product(e_2, e_1, e_1))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(223,plain,
% 0.19/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | (~product(Z2, Y, X)) | product(X, Y, Z1))) | product(e_2, e_1, e_1) | (~product(e_2, e_1, e_2))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[222, 221])).
% 0.19/0.43 tff(224,plain,
% 0.19/0.43 (product(e_2, e_1, e_1)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[223, 51, 217])).
% 0.19/0.43 tff(225,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_2) | (~product(e_2, e_1, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | product(e_1, e_1, e_2) | (~product(e_2, e_1, e_1)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(226,plain,
% 0.19/0.43 (((~product(e_2, e_1, e_1)) | product(e_1, e_1, e_2) | (~product(e_2, e_1, e_1))) <=> (product(e_1, e_1, e_2) | (~product(e_2, e_1, e_1)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(227,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | ((~product(e_2, e_1, e_1)) | product(e_1, e_1, e_2) | (~product(e_2, e_1, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | (product(e_1, e_1, e_2) | (~product(e_2, e_1, e_1))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[226])).
% 0.19/0.43 tff(228,plain,
% 0.19/0.43 (((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | ((~product(e_2, e_1, e_1)) | product(e_1, e_1, e_2) | (~product(e_2, e_1, e_1)))) <=> ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | product(e_1, e_1, e_2) | (~product(e_2, e_1, e_1)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[227, 225])).
% 0.19/0.43 tff(229,plain,
% 0.19/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | ((~product(e_2, e_1, e_1)) | product(e_1, e_1, e_2) | (~product(e_2, e_1, e_1)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(230,plain,
% 0.19/0.43 ((~![Y: $i, Z1: $i, X: $i, Z2: $i] : ((~product(X, Z1, Z2)) | product(Z2, Y, X) | (~product(X, Y, Z1)))) | product(e_1, e_1, e_2) | (~product(e_2, e_1, e_1))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[229, 228])).
% 0.19/0.43 tff(231,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[230, 172, 104, 224])).
% 0.19/0.43 % SZS output end Proof
% 0.19/0.44 % E exiting
%------------------------------------------------------------------------------